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Instantons and jumping lines. (English) Zbl 0596.32031
The author investigates the moduli space of SV(2)-instantons of topological charge n. By means of the Ward transform this is equivalent to the space of certain holomorphic bundles on $${\mathbb{C}}{\mathbb{P}}_ 3$$ and these bundles may be described by monads. Recently, S. Donaldson [Commun. Math. Phys. 93, 453-460 (1984; Zbl 0581.14008)] has identified this space of monads with the moduli space of semi-stable rank 2 holomorphic vector bundles on $${\mathbb{P}}_ 2$$. Blowing up one point in $${\mathbb{P}}_ 2$$ gives a ruled surface on which the pull-back of a semi- stable bundle will jump at n lines of the ruling. The author carefully analyzes this jumping behaviour so that one may reconstruct the given bundle from suitable jumping data at these distinguished lines. This gives him enough information to be able to calculate the fundamental group of the original moduli space: $${\mathbb{Z}}_ 2$$ for n even and 0 for n odd. The moduli space of stable bundles is much easier to work with than the monad description especially because the reality conditions have disappeared. This paper is a good illustration of this fact. This was also Donaldson’s motivation of course (which immediately allowed him to conclude connectivity of the moduli spaces).
Reviewer: M.Eastwood

##### MSC:
 32G05 Deformations of complex structures 32L05 Holomorphic bundles and generalizations 53C80 Applications of global differential geometry to the sciences 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 32L25 Twistor theory, double fibrations (complex-analytic aspects) 57S30 Discontinuous groups of transformations
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##### References:
 [1] Atiyah, M.F.: The geometry of Yang-Mills fields. Accademia Nazionale dei Lincei, Scuola Normale Superiore, Pisa 1979 · Zbl 0435.58001 [2] Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys.93, 437-451 (1984) · Zbl 0564.58040 · doi:10.1007/BF01212288 [3] Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Yu.I.: Construction of instantons. Phys. Lett.65A, 185-187 (1978) · Zbl 0424.14004 [4] Barth, W.: Some properties of stable rank-2 vector bundles on ? n . Math. Ann.226, 125-150 (1977) · Zbl 0417.32013 · doi:10.1007/BF01360864 [5] Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math.42, 63-91 (1977) · Zbl 0386.14005 · doi:10.1007/BF01389784 [6] Donaldson, S.K.: Instantons and geometric invariant theory. Commun. Math. Phys.93, 453-461 (1984) · Zbl 0581.14008 · doi:10.1007/BF01212289 [7] Grauert, H., Mülich, G.: Vektorbündel vom Rang 2 über demn-dimensionalem komplex projektiven Raum. Manuscr. Math.16, 75-100 (1975) · Zbl 0318.32027 · doi:10.1007/BF01169064 [8] Grothendieck, A.: Eléments de géométrie algébrique. III. Publ. Math. IHES [9] Grothendieck, A.: Sur la classification des fibres holomorphes sur la sphère de Riemann. Am. J. Math.79, 121-138 (1957) · Zbl 0079.17001 · doi:10.2307/2372388 [10] Hartshorne, R.: Algebraic geometry. Berlin, Heidelberg, New York: Springer GTM (1978) · Zbl 0367.14001 [11] Hartshorne, R.: Stable vector bundles and instantons. Commun. Math. Phys.59, 1-15 (1978) · Zbl 0383.14006 · doi:10.1007/BF01614151 [12] Horrocks, G.: Vector bundles on the punctured spectrum of a local ring. Proc. Lond. Math. Soc. (3)14, 689-713 (1964) · Zbl 0126.16801 · doi:10.1112/plms/s3-14.4.689 [13] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Prog. Math. 3. Boston: Birkhäuser 1980 · Zbl 0438.32016 [14] Le Potier, J.: Fibres stables de rang 2 sur ?2(?). Math. Ann.241, 217-256 (1979) · Zbl 0405.14008 · doi:10.1007/BF01421207
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